The positions of four of the smaller corner pieces depend on the positions of the other 4 corner pieces, and only even permutations of these positions are possible.
12.
An even permutation is one that can be represented by an even number of swaps while an odd permutation is one that can be represented by an odd number of swaps.
13.
Taking the even permutations with an odd number of plus signs, and the odd permutations with an even number of plus signs, gives a different snub cube, the mirror image.
14.
There are 4 6 / 2 ( 2, 048 ) ways to orient the centres, since an even permutation of the corners implies an even number of quarter turns of centres as well.
15.
It has the following two normal subgroups, the group of even permutations on the corners " A " 8 and the group of even permutations on the edges " A " 12.
16.
It has the following two normal subgroups, the group of even permutations on the corners " A " 8 and the group of even permutations on the edges " A " 12.
17.
The first constraint is that only even permutations of the face centers are possible ( e . g . it is impossible to have only two face centre pieces swapped ); this divides the limit by 2.
18.
Though it may not be immediately obvious, the figure obtained by taking the even permutations with an even number of plus signs is the same as that obtained by taking the odd permutations with an odd number of plus signs.
19.
By defining the parity of \ sigma as the parity of N ( \ sigma ), a permutation that has an even length decomposition is an even permutation and a permutation that has one odd length decomposition is an odd permutation.
20.
The positions of four of the face centers is completely determined by the positions of the other 4 face centers, and only even permutations of such positions are possible, so the number of arrangements of face centers is only 4 ! / 2.
